Polar Form

3 Key excerpts on "Polar Form"

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  eBook - ePub

  Barron's Math 360: A Complete Study Guide to Pre-Calculus with Online Practice

  • Lawrence S. Leff, Christina Pawlowski-Polanish(Authors)
  • 2021(Publication Date)
  • Barrons Educational Services

    (Publisher)

  Figure 13.11 . This is known as a limaçon.

  FIGURE 13.11

  Below is a table of common polar graphs, where a and b are constants.

  Lesson 13-3: The Polar Form of a Complex Number

  KEY IDEAS

  When complex numbers of the form a + bi are graphed, the rectangular coordinates (a,b) are plotted where a is measured along the horizontal real axis and b is measured along the vertical imaginary axis. The set of all such points forms the complex plane.

  In Lesson 13-2 , we saw that representing the same point in polar and rectangular coordinate systems produced formulas that allowed us to convert from one system to the other. In a similar way, superimposing the complex plane on the plane created by the polar coordinate system produces formulas that allow us to convert a complex number into Polar Form.

  POLAR REPRESENTATION OF A COMPLEX NUMBER

  Let point (a,b) in the complex plane correspond to the complex number a + bi, and let (r,θ) represent the polar coordinates of the same point, as illustrated in Figure 13.12 . Using the familiar relationships a = r cos θ and b = r sin θ that derive from the right triangle in this figure, we can write the complex number a + bi in Polar Form as

  FIGURE 13.12 Complex and polar coordinate systems

  Polar Form OF A COMPLEX NUMBER

  The Polar Form of the complex number z = a + bi is

  z = r (cos θ + i sin θ),

  where r2 = a2 + b2 and tan . The r-value is called the modulus of z, and θ is termed an argument of z.

  The Polar Form of the complex number, , is sometimes referred to as the trigonometric form or the “rcis” form of a complex number. For example, the shorthand notation 3 cis means .

  EXERCISE 1 Converting Complex Numbers from Rectangular to Polar Form Convert each complex number into Polar Form.

  a.

  b.5 + 0i

  c.0 – 4i

  SOLUTIONS

  a.To write in Polar Form, find θ and r when a = 1 and .

  •Find the reference angle:

  •Determine θ. Because a is positive and b is negative in Quadrant IV, θ

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  eBook - ePub

  Protective Relaying

  Principles and Applications, Fourth Edition

  • J. Lewis Blackburn, Thomas J. Domin(Authors)
  • 2014(Publication Date)
  • CRC Press

    (Publisher)

  Chapter 3

  Phasors and Polarity

  3.1 Introduction

  Phasors and polarity are two important and useful tools in power system protection. They aid in understanding and analyzing the connections, operation, and testing of relays and relay systems. In addition, these concepts are essential in understanding power system performance during both normal and abnormal operation. Thus, a sound theoretical and practical knowledge of phasors and polarity is a fundamental and valuable resource.

  3.2 Phasors

  The IEEE Dictionary (IEEE 100) defines a phasor as “a complex number.” Unless otherwise specified, it is used only within the context of steady-state alternating linear systems. It continues: “the absolute value (modulus) of the complex number corresponds to either the peak amplitude or root-mean-square (rms) value of the quantity, and the phase (argument) to the phase angle at zero time. By extension, the term ‘phasor’ can also be applied to impedance, and related complex quantities that are not time dependent.”

  In this book, phasors will be used to document various AC voltages, currents, fluxes, impedances, and power. For many years phasors were referred to as vectors , but this use is discouraged to avoid confusion with space vectors. However, the former use lingers on, so occasionally a lapse to vectors may occur.

  3.2.1 Phasor Representation

  The common pictorial form for representing electrical and magnetic phasor quantities uses the Cartesian coordinates with x (the abscissa) as the axis of real quantities and y (the ordinate) as the axis of imaginary quantities. This is illustrated in Figure 3.1 . Thus, a point c on the complex plane x –y

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  eBook - ePub

  Mathematical Economics

  • Arsen Melkumian(Author)
  • 2012(Publication Date)
  • Routledge

    (Publisher)

  z

  Note that compex conjugates have the same modulus: EXAMPLE 11.4

  (a) |3 + 4i| = 5

  (b) |3 − 4i| = 5

  (c)

  (d)

  Using complex numbers we can solve any quadratic equation. Recall that the solution for ax2 + bx + c = 0 is given by

  where

  We can use the formula over the complex numbers even when the discriminant D is negative.

  EXAMPLE 11.5 Solve

  Solution:

  EXAMPLE 11.6 Prove that .

  Solution: Let z1 = a + bi and z2 = c + di. Then

  EXAMPLE 11.7 Show that .

  Solution: Let z1 = a + bi and z2 = c + di. Then

  11.2 Polar and trigonometric form of complex numbers

  So far we have discussed the so-called Cartesian representation of the complex numbers. In essence, we have assigned to each complex number a point on the Cartesian (or complex) plane. However, complex numbers have also a representation in polar coordinates. A complex number z = a + bi is assigned coordinates (r, θ) in the polar system, where

  and θ is such that

  as shown in Figure 11.3 .

  The pair (r, θ) specifies a unique point on the complex plane. However, a given point on the complex plane does not have a unique polar representation. In fact, a point z = (r, θ) has infinitely many polar representations of the form z = (r, θ + 2πn), with n ∈ ℤ. A number in the form (θ + 2πn) is called an argument of z. The argument of z lying in the range (−π, π] is referred to as the principal argument of z or Arg(z). For instance, the principal argument of 2i is and the principal argument of (2i + 2) is

  Figure 11.3

  A complex number z can also be written in a trigonometric form:

  The trigonometric form of complex numbers is used extensively in time-series analysis.

  EXAMPLE 11.8 Find the trigonometric form of z = 2 + 2i.

  Solution:

  It follows that Hence Complex numbers can also be written in an exponential form. Euler’s formula from complex analysis states that

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